I totally agree with Anthony about the Feelworld, but if you want something even brighter for outdoor shoots, you should check out the Desview R6 UHB 5.5 Inch Field Monitor. Its like 2800 nits which is way better for seeing things in direct sunlight compared to the standard F5 Pro. Waveforms are non-negotiable imo, especially since the a7 III screen is so small and hard to judge exposure on. One thing to keep in mind is power. These monitors drain batteries fast. I usually pick up a couple of Neewer NP-F970 6600mAh Li-ion Batteries so i dont have to worry about it dying mid-shoot. Also, definitely get a dedicated tilt arm or a small cage mount instead of just a cold shoe ball head... those cheap ball heads always slip and it gets annoying really fast.

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The FEELWORLD F5 Pro is a popular sub-$200 option often paired with Sony cameras (Source 1). Prioritize waveform/histogram displays for accurate exposure. While hotshoe monitors exist, a field monitor with its own battery offers more flexibility (like the F5 Pro). Don't expect top-tier color, but it's a solid budget choice.